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2-d: images

Let us assume we have a continuous distribution, on a plane, of values of luminance or, more simply stated, animage. In order to process it using a computer we have to reduce it to a sequence of numbers by means ofsampling. There are several ways to sample an image, or read its values of luminance at discrete points. Thesimplest way is to use a regular grid, with spatial steps X e Y . Similarly to what we did for sounds, we define the spatial sampling rates F X 1 X and F Y 1 Y . As in the one-dimensional case, also for two-dimensional signals, or images, sampling can bedescribed by three facts and a theorem.

  • The Fourier Transform of a discrete-space signal is a function (called spectrum ) of two continuous variables ω X and ω Y , and it is periodic in two dimensions with periods 2 π . Given a couple of values ω X and ω Y , the Fourier transform gives back a complex number that can be interpreted as magnitude andphase (translation in space) of the sinusoidal component at such spatial frequencies.
  • Sampling the continuous-space signal s x y with the regular grid of steps X , Y , gives a discrete-space signal s m n s m X n Y , which is a function of the discrete variables m and n .
  • Sampling a continuous-space signal with spatial frequencies F X and F Y gives a discrete-space signal whose spectrum is the periodic replication along the grid of steps F X and F Y of the original signal spectrum. The Fourier variables ω X and ω Y correspond to the frequencies (in cycles per meter) represented by the variables f X ω X 2 π X and f Y ω Y 2 π Y .

The [link] shows an example of spectrum of a two-dimensional sampled signal. There, thecontinuous-space signal had all and only the frequency components included in the central hexagon. The hexagonalshape of the spectral support (region of non-null spectral energy) is merely illustrative. The replicas of the originalspectrum are often called spectral images .

Spectrum of a sampled image

Given the above facts , we can have an intuitive understanding of the Sampling Theorem.

Sampling theorem (in 2d)

A continuous-space signal s x y , whose spectral content is limited to spatial frequencies belonging to the rectangle having semi-edges F bX and F bY (i.e., bandlimited) can be recovered from its sampled version s m n if the spatial sampling rates are larger than twice the respective bandwidths (i.e., if F X 2 F bX and F Y 2 F bY )

In practice, the spatial sampling step can not be larger than the semi-period of the finest spatial frequency (or thefinest detail) that is represented in the image. The reconstruction can only be done through a filter thateliminates all the spectral images but the one coming directly from the original continuous-space signal. In otherwords, the filter will cut all images whose frequency components are higher than the Nyquist frequency defined as F X 2 and F Y 2 along the two axes. The condition required by the sampling theorem is equivalent to requiring that there are no overlaps between spectral images. If there were suchoverlaps, it wouldn't be possible to eliminate the copies of the original signal spectrum by means of filtering. In case of overlapping, a filtercutting all frequency components higher than the Nyquist frequency would give back a signal that is affected byaliasing.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Media processing in processing. OpenStax CNX. Nov 10, 2010 Download for free at http://cnx.org/content/col10268/1.14
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